Heat content and horizontal mean curvature on the Heisenberg group

TL;DR

This paper derives the short-time asymptotics of the heat content in the Heisenberg group, linking it to geometric boundary measures, and extends classical results to the sub-Riemannian setting.

Contribution

It generalizes previous heat content asymptotics to the sub-Riemannian context of the Heisenberg group, relating coefficients to horizontal perimeter and mean curvature.

Findings

Asymptotic formula for heat content in the Heisenberg group

Coefficients expressed via horizontal perimeter and mean curvature

Probabilistic proof using Brownian motion

Abstract

We identify the short time asymptotics of the sub-Riemannian heat content for a smoothly bounded domain in the first Heisenberg group. Our asymptotic formula generalizes prior work by van den Berg-Le Gall and van den Berg-Gilkey to the sub-Riemannian context, and identifies the first few coefficients in the sub-Riemannian heat content in terms of the horizontal perimeter and the total horizontal mean curvature of the boundary. The proof is probabilistic, and relies on a characterization of the heat content in terms of Brownian motion.